What Arkani-Hamed and his collaborators proposed was that a similar argument would suggest that if gravitational forces propagated in extra dimensions, as well as in our three-dimensional space, then the strength of the gravitational force measured between massive particles in our space would fall off faster than the inverse of the square of the distance between them. Imagine, for example, a single extra dimension. If field lines could spread out in our three dimensions, but also in this extra dimension, then the number of field lines per unit area would fall off as the area of a threedimensional spherical surface (bounding a four-dimensional volume), and not as the two-dimensional spherical surface bounding a three-dimensional volume that we normally picture when we draw field lines spreading out into space. Since the area of a three-dimensional spherical surface increases with the cube of its radius, and not the square of its radius, as in a twodimensional spherical surface, this means that the strength of gravity would fall off inversely with the cube of distance, not the square of distance. There is, of course, a slight problem here. Newton achieved fame and fortune by demonstrating that a universal gravitational force that fell off with the square of distance could explain everything from falling apples to the orbits of planets! So, what gives?
Well, it is true that gravity has been measured with great precision to have an inverse square law on scales ranging from human scales to galactic scales. But, as Arkani-Hamed and collaborators pointed out, it hadn’t been so measured at scales smaller than about a millimeter. Imagine, then, that the extra spatial dimension has a size of a millimeter. Then for objects separated in our space by less than about a millimeter, the force of gravity will fall off with the cube of distance. But once objects get separated by a larger amount than this, the gravitational field lines from one particle cannot spread out any more in the extra dimension as they can do in our three dimensions of space. As the field lines can continue to spread out only in the three remaining large dimensions on scales larger than a millimeter, the gravitational field again begins to now fall off inversely with the square of distance. An example starting with three dimensions, one of which is small, is shown below:
What is the net effect of all of this? Well, if the size of the extra dimension is R, then gravity falls off with one extra power of distance between the Planck scale and R, compared to what it would be if there were only three dimensions for gravity to propagate in. Thus, by the time objects are separated by a distance R, the gravitational attraction between them would be weaker by a much larger factor than they would otherwise have been. If one is only measuring gravity on scales larger than R, gravity would then be measured to fall off with an inverse square law, just as Newton argued, but by this time the apparent strength of gravity would be much smaller than it would have been if the gravitational field had not been able to fall, at least temporarily, faster than inversely with the square of distance. Without knowledge of these extra dimensions, this extra suppression factor would simply be incorporated into the basic definition of the strength of gravity itself. This strength is given by what we conventionally call Newton’s constant, which appears in the formula for the inverse square law gravitational force between two bodies.
By a similar argument, if there are two extra hidden compactified dimensions, then gravity will fall off by an even greater factor between the real Planck scale and the scale, R, of the extra dimensions, and so on for yet more compactified dimensions.
Because it tells us how strong the gravitational force is between measured bodies, we use Newton’s constant to determine on what distance or energy scale we expect that quantum mechanical gravitational effects should become significant. Thus, it is Newton’s constant that determines the value of what we conventionally define as the Planck scale. As a result, because of the hidden effects of the extra dimensions on scales between the Planck scale R, we would “inaccurately” deduce the Planck scale to be much higher than it actually is. This is because the strength of the gravitational force would grow much faster as one decreases the distance between massive objects, on scales smaller than R, than we would otherwise have expected. What Arkani-Hamed and collaborators realized is that this would allow the possibility that the real Planck scale might actually be equal to the distance scale at the point where the weak and electromagnetic interactions are unified, instead of seventeen orders of magnitude smaller, as we would otherwise estimate based on our incorrect extrapolation of the behavior of gravity on scales smaller than R. This would therefore naturally explain the apparent large hierarchy between the electroweak scale and the Planck scale. The hierarchy problem would therefore be a problem of our own making; no such actual hierarchy would exist in nature. There is an immediate concern, however. With just one extra dimension, the extra falloff in the strength of gravity from the real Planck scale to the size of the extra dimension, R, is sufficiently slow so that to yield the strength of gravity that we actually do measure on large scales forces this latter size R to be roughly equal to the size of our solar system! This is clearly impossible, since this would mean that gravity would be measured to fall off inversely with the cube of distance throughout our solar system, when it was precisely the measurements of gravity over our solar system that led Newton to propose the inverse square law in the first place. Clearly then, one such extra large dimension is ruled out by observation. However, if there were two extra new dimensions, then because gravity would fall off even faster with distance for distances smaller than R, this would allow the size of the extra dimension, R, to be much smaller than it would be in the case of one extra dimension. If one works out the numbers, the scale R would only have to be about one millimeter. In this case, the extra dimensions would indeed be as large as small pebbles. Not only is such a possibility not ruled out, but this is precisely the scale at which new experiments had been designed to explore the inverse square law behavior of gravity. If Arkani-Hamed and collaborators were correct, and if there are only two extra large dimensions, these experiments could measure something quite different, revealing for the first time the hidden dimensions that have otherwise remained within the realm of theorists’ imaginations for all these years.
To recap: It is perfectly possible for extra dimensions to exist and be relatively large, provided that the only force that can propagate in these extra dimensions is gravity. Moreover, one can resolve the hierarchy problem for gravity, making the Planck scale, and with it presumably the string scale, essentially equal to the electroweak scale if there are two extra large dimensions into which gravity can propagate, both of which are about a millimeter in size.
As I have indicated, the possibility that extra dimensions, such as those that might be associated with string theory, might be large enough to actually be measured sent a jolt of excitement through much of the particle physics community that was perhaps stronger than any that had been experienced since the first string revolution of 1984. Suddenly a host of potential new experimental probes—not only of quantum gravity, but also of string theory and even extra dimensions—would become feasible. One of the most exciting such exotic probes involves exploring strings and extra dimensions at current or planned particle physics accelerators. For if the Planck scale and with it the string scale coincide with the electroweak scale, then machines designed to explore weak interaction physics could uncover exotic new phenomena. Higher-energy string excitations in extra dimensions might be excited in high-energy particle collisions, which would be manifested in precisely the same tower of new particle states (albeit at now much higher energies) that had first been predicted when strings had been proposed as a theory of the strong interactions. Equally interesting would be the possibility that some of the energy in these highenergy collisions might literally disappear in gravitational waves that could move off into the extra dimensions.
Finally, perhaps the most exciting prediction of all would be that gravity itself would become strong enough at the electroweak scale so that new quantum gravitational phenomena might be directly observable there.
For example, high-energy collisions in new accelerators might produce primordial, elementary, particlelike “black holes,” which might then spontaneously decay in a burst of radiation, as predicted by Hawking. Not only would such new signatures be striking, they would allow us to confirm one of the key phenomena predicted to occur when quantum mechanical effects are incorporated into gravity and ultimately directly explore one of our most puzzling paradoxes, the information loss paradox. Any of these experiments might be exciting enough to get one’s juices flowing, even if they are long shots, but for those who truly crave dimensions large enough to hide aliens in, millimeter sizes, even if huge by comparison to what had previously been assumed in string theory, just don’t cut it.
Happily an even more exotic possibility was independently proposed within a year of Arkani-Hamed and coworkers’ theory, by Lisa Randall, now at Harvard University, and a past student of mine, Raman Sundrum, currently at Johns Hopkins.
Randall and Sundrum argued that there is another way to resolve the hierarchy problem using extra dimensions that is quite distinct, and certainly more subtle, than the mechanism proposed by Arkani-Hamed and colleagues. They proposed starting with a single compact extra dimension, but not one completely independent of our own. In the true spirit of Star Trek they introduced what they called a “warp factor,” though theirs has nothing to do with faster-than-light travel. Rather, it arises from the suggestion that an extra dimension exists that is strongly curved (or “warped”) as one moves away from the three-brane that makes up the three-dimensional world we experience.
What Randall and Sundrum realized is that, in this case, even if the size of the extra compactified dimension is perhaps only of order of ten to fifty times larger than the Planck scale, it is still possible to produce a natural large hierarchy, of perhaps fifteen orders of magnitude, between this scale and the scale of the elementary particle masses and interactions we observe.
The effect is subtle and somewhat difficult to directly picture physically without recourse to mathematics, as is due to effects of curvature in the extra dimension. Remember that general relativity tells us that the curvature of space is related to the overall magnitude of the mass and energy of objects within the space. Now the curvature associated with the warping of the extra dimension near our three-brane, in the Randall-Sundrum picture, could be rather large, characteristic of energies near the Planck energy scale. But, if the extra dimension is perhaps fifty times larger than the characteristic scale over which it curves, then when one solves the full five-dimensional equations associated with general relativity, a hierarchy appears. It turns out that even if, in the five-dimensional theory, the fundamental mass and energy parameters are all of the order of the Planck scale, in our four-dimensional world all fundamental particle masses will instead appear to be suppressed compared to the Planck scale by a factor of 1015.
Randall and Sundrum also showed that there was another slightly more intuitive way of thinking of this problem, in terms of the relative strength of the forces in nature. The hierarchy problem can be recast as follows: Gravity is measured to be more than a billion billion billion billion times weaker than electromagnetism, and even weaker still when compared to the strong force. It may not seem so weak, especially in the morning when you try and pull yourself out of bed, but remember that you are feeling the gravitational force of the entire earth acting on you. By contrast, even a small excess of electric charge on an object such as a balloon produces a large enough electric field to hold it up on a wall against the gravitational pull of the entire earth. The hierarchy problem involves the question of why there is this huge discrepancy. In the Randall-Sundrum scheme, the warping of space near our threebrane implies that gravity near our brane acts effectively much more weakly than it does outside our brane. The exponential warping, in fact, makes gravity appear exponentially weaker on our brane than it is at the other side of five-dimensional space. If we happened to live on a threebrane located there, which I remind you is located merely a microscopic distance “away” from our world in the extra spatial dimension, gravity would appear to have the same strength as the other forces in nature. The observed hierarchy in our world then becomes merely an environmental accident. Gravity “leaks” into our dimension as surely as Buckaroo Banzai’s extra-dimensional nemesis does. Like the Arkani and coworkers scenario, Randall-Sundrum’s extradimensional solution of the hierarchy problem would bring extra dimensions into the realm of the testable. In this case, only the massless particle (the graviton) that conveys the gravitational force would be weakly coupled on our brane. As in all compactified theories, there would also be a tower of higher-mass particles that could be produced if one had sufficient energy. In the Randall-Sundrum model, however, these higher-dimensional gravitational modes would have masses characteristic of the electroweak scale and coupling strengths not characteristic of gravity, but rather of electroweak physics. The new extra-dimensional states would thus be produced in great abundance, with observable decay modes, just like ordinary particles, if one had an accelerator that could achieve the necessary energies. And, interestingly, just such an accelerator is being built at CERN (the European Center for Nuclear Research, in Geneva, Switzerland) and is due to come online in 2007 or 2008.
In fact, not only would new higher-dimensional gravitational excitations be produced at such an accelerator if this idea is correct, but at slightly higher scales fundamental strings could also be generated and explored. All the mysteries of string theory or M-theory would be laid bare for experimentalists to probe, even if theorists have remained, by that time, unable to untangle their complexities.
Now, before you go out and buy CERN futures, you might want to step back and note a few of the hidden, but profound, problems with this model as it stands. First and foremost is an issue that has plagued all Kaluza-Klein theories since their origin: Why are the extra dimensions small, and our three-brane possibly infinitely large? There has simply been no good answer to this question in the past ninety years. While a great deal of work has been devoted to trying to find physical mechanisms that would allow such a possibility, no real progress has been made. It is simply assumed that something happens so that the extra dimensions remain hidden, whether they curl up on the size of the Planck scale or are as large as a pebble on the road.
Actually, the situation is often even worse than I have thus far described. In general, it turns out that the dynamic equations of the theory tend to drive the size of the extra dimension to be infinitely large, as presumably are the three spatial dimensions in which we live, even if one initially starts the extra dimensions off to be the Planck scale, say. This embarrassment is solved in the way other similar confusing aspects of string theory and M-theory are sometimes dealt with: Namely, it is assumed that when we fully understand the ultimate theory, everything will become clear.
Nevertheless, aside from this “minor” inadequacy, you may recall that I promised you a theory with a really large extra dimension, not the puny compactified extra dimension that Randall and Sundrum proposed. Happily, for those who find the unbridled optimism of the last paragraph less than convincing, these researchers discovered, within a month of their original suggestion, that a compactified extra dimension was, in fact, completely unnecessary in their warped five-dimensional space-time model. If the space outside our local three-brane was warped, they discovered that the size of the extra dimension(s) could in fact be infinite . . . namely, just as big as the three dimensions of our experience!
To reach this conclusion, they considered—instead of a fivedimensional space with a compactified extra dimension—an infinitely large five-dimensional space with two three-branes located a very small distance apart. The finite volume between the branes mimicked the compactified space of their original model. They then assumed that we live on one of the two three-branes, and considered what happens as one slowly increases the separation of the two branes.
You will recall that in their original scenario, the warping of spac
e-time near our three-brane caused the strength of gravity to fall off exponentially as one approached our brane from anywhere else in the space. In this case, they switched things around, having the strength of gravity fall off exponentially away from our brane. One finds, accordingly, that the force of gravity is effectively tied to our brane and, since gravity is our only probe of the extra dimension, even in the limit that the other brane is removed to infinite distances in the fifth dimension the effects of this large extra dimension are completely hidden. Also, it turns out that the masses of the tower of states that occur in higher dimensional theories all tend toward zero, just like the (zero-mass) graviton that is responsible for conveying the gravitational force in our three-dimensional space. But, fortunately all these extra states essentially decouple from matter in our space, and therefore cannot be produced in any present or planned accelerators, and remain completely unobservable. Moreover, any corrections they might provide to the nature of the gravitational force between test particles is suppressed by the ratio of the distance between the test particles divided by the curvature scale in the extra dimension. If this latter quantity is on the order of the Planck scale, then the effect will be unobservably small in any conceivable experiment that might be performed in the future. What this second Randall-Sundrum model demonstrates is that the conventional wisdom about extra dimensions, stretching back all the way to Kaluza-Klein, was wrong. It is completely possible to hide behind the mirror not only a microscopic extra dimension, as originally envisaged, or even merely a tiny extra dimension, as Arkani-Hamed and colleagues envisaged, but also an infinite extra dimension, which would exist in concert with our own three-dimensional space.
Hiding in the Mirror: The Quest for Alternate Realities, From Plato to String Theory (By Way of Alicein Wonderland, Einstein, and the Twilight Zone) Page 24
